Onset of Finger Convection in a Horizontal Porous Layer Underlying a Fluid Layer

In the directional solidification of concentrated alloys, the frozen solid region is separated from the melt region by a mushy zone consisting of dendrites immersed in the melt. Simultaneous occurrence of temperature and solute gradients through the melt and mushy zones may be conducive to the occurrence of salt-finger convection, which may in turn cause adverse effects such as channel segregation. We have considered the problem of the onset of finger convection in a porous layer underlying a fluid layer using linear stability analysis. The eigenvalue problem is solved by a shooting method. As a check on the method of solution and the associated computer program, we first consider the thermal convection problem. In this process, it is discovered that at low depth ratios dˆ (the ratio of the fluid layer depth to the porous layer depth), the marginal stability curve is bimodal. At small dˆ, the long-wave branch is the most unstable and the convection is dominated by the porous layer. At large dˆ, the short-wave branch is the most unstable and the convection is dominated by the fluid layer, with a convection pattern consisting of square cells in the fluid layer. In the salt-finger case with a given thermal Rayleigh number Ra_m = 50, as the depth ratio dˆ is increased from zero, the critical salt Rayleigh number Ra_sm first decreases, reaches a minimum, and then increases. The system is more stable at dˆ > 0.2 than at dˆ = 0. This rather unusual behavior is again due to the fact that at small dˆ, convection is dominated by the porous layer and, at large dˆ, convection is dominated by the fluid layer. However, in the latter case, the convection pattern in the fluid layer consists of a number of high aspect ratio cells.